TITLE: The generalization of Brouwer's fixed point theorem?
QUESTION [8 upvotes]: Let $X$ be a contractible compact [edit: locally connected] topological space 
(Hausdorff and second countable). Let $f\colon X\to X$
be a continuous map. Then (I suppose) $f$ has a fixed 
point. Personally, I cannot think of a better generalization 
of Brouwer's fixed point theorem, but is it true?

REPLY [24 votes]: No.  I believe the first counterexample is from:
Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 96-98
which I unfortunately can't find online.  Kinoshita's example is described on page 127 in this excellent article by Bing, however.
EDIT:  The question has been revised to add the local connectivity condition; as stated, I think the question is open.  If "contractible" is replaced with "acyclic" there are counterexamples dating back to Borsuk, referenced e.g. here; Borsuk's paper, which I can't find online, is:
K. Borsuk, Sur un continua acyclique qui se laisse transformer topologiquement en lui-meme sans points invariants, Fund. Math. 24 (1935), pp. 51-58.
This suggests that if the OP's conjecture is true, it is unlikely to be open to homological attacks.